Borwein Integrals: A Pattern That Breaks at Step 8
∫sinc(x)·sinc(x/3)·sinc(x/5)·…dx = π for steps 1–7. At step 8 it silently breaks.
∫ sinc(x) dx = π
Step k
Factor added
Running sum S_k
Integral value
= π ?
Running Sum S_k vs 1.0 threshold
k=1k=12
Red line = S_k = 1.0
Bar turns red when S_k crosses 1.0 (at k=8)
Bonus — with 2cos(x) modifier:
The pattern holds even longer. The break occurs when S_k = 1/3+1/5+…+1/(2k-1) crosses 2.0, which happens at sinc(x/113) (step k=57).